{"id":5216,"date":"2014-04-19T06:05:46","date_gmt":"2014-04-18T21:05:46","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5216"},"modified":"2014-08-01T18:35:03","modified_gmt":"2014-08-01T09:35:03","slug":"determinants","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5216","title":{"rendered":"Determinants"},"content":{"rendered":"
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If the matrix A in (1) is a square matrix, then we associate with A a number denoted by <\/p>\n

\\displaystyle \\Delta = \\left| \\begin{array}{cccc} a_{11} & a_{12} & \\cdots & a_{an} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{array} \\right|\\cdots (9)<\/p>\n

called the determinant<\/em> of A of order n<\/em>, written det(A). In order to define the value of a determinant, we introduce the following concepts. <\/p>\n

1. Minor<\/h4>\n

Given any element a_{jk} of \\Delta we associate a new determinant of order (n – 1) obtained by removing all elements of the j<\/em>th row and k<\/em>th column called the minor of a_{jk}. <\/p>\n

2. Cofactor<\/h4>\n

If we multiply the minor of a_{jk} by (-1)^{j+k}, the result of the elements in any row [or column] by their corresponding cofactors and is called the Laplace expansion<\/em>. In symbols, <\/p>\n

\\displaystyle \\det{A} = \\sum^{n}_{k=1}a_{jk}A_{jk} \\cdots (10)<\/p>\n

We can show that this value is independent of the row [or column] used. <\/p>\n<\/blockquote>\n